Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. See Examples 1–4. ƒ(x) = -3x^2 + 24x - 46

Verified step by step guidance

1. Identify the quadratic function given: \(f(x) = -3x^2 + 24x - 46\).
2. Find the vertex using the vertex formula. The x-coordinate of the vertex is given by \(x = -\frac{b}{2a}\), where \(a = -3\) and \(b = 24\).
3. Calculate the y-coordinate of the vertex by substituting the x-value found into the original function: \(f(x) = -3x^2 + 24x - 46\).
4. Determine the axis of symmetry, which is the vertical line passing through the vertex, given by the equation \(x = \text{(x-coordinate of vertex)}\).
5. State the domain and range: The domain of any quadratic function is all real numbers, \((-\infty, \infty)\). Since \(a = -3\) is negative, the parabola opens downward, so the range is \((-\infty, \text{y-coordinate of vertex}]\).

Key Concepts

Vertex of a Quadratic Function

The vertex is the highest or lowest point on the graph of a quadratic function, representing its maximum or minimum value. It can be found using the formula x = -b/(2a) for a function in standard form f(x) = ax^2 + bx + c. Substituting this x-value back into the function gives the y-coordinate of the vertex.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror-image halves. Its equation is x = -b/(2a), the same x-value used to find the vertex. This line helps in graphing and understanding the symmetry of the quadratic function.

Domain and Range of Quadratic Functions

The domain of any quadratic function is all real numbers since x can take any value. The range depends on the direction of the parabola: if it opens upward (a > 0), the range is all y-values greater than or equal to the vertex's y-coordinate; if it opens downward (a < 0), the range is all y-values less than or equal to the vertex's y-coordinate.